Design and optimization of dynamic reliability-driven order allocation and inventory management decision model

Efficient order allocation and inventory management are essential for the success of supply chain operations in today’s dynamic and competitive business environment. This research introduces an innovative decision-making model incorporating dependability factors into redesigning and optimizing order allocation and inventory management systems. The proposed model aims to enhance the overall reliability of supply chain operations by integrating stochastic factors such as demand fluctuations, lead time uncertainty, and variable supplier performance. The system, named Dynamic Reliability-Driven Order Allocation and Inventory Management (DROAIM), combines stochastic models, reliability-based supplier evaluation, dynamic algorithms, and real-time analytics to create a robust and flexible framework for supply chain operations. It evaluates the dependability of suppliers, transportation networks, and internal procedures, offering a comprehensive approach to managing supply chain operations. A case study and simulations were conducted to assess the efficacy of the proposed approach. The findings demonstrate significant improvements in the overall reliability of supply chain operations, reduced stockout occurrences, and optimized inventory levels. Additionally, the model shows adaptability to various industry-specific challenges, making it a versatile tool for practitioners aiming to enhance their supply chain resilience. Ultimately, this research contributes to existing knowledge by providing a thorough decision-making framework incorporating dependability factors into order allocation and inventory management processes. Practitioners and experts can implement this framework to address uncertainties in their operations.


INTRODUCTION
In today's rapidly evolving and fiercely competitive business landscape, the efficacy of supply chain operations stands as a linchpin for organizational success (Hajar & Saida, 2022).The intricate interplay of factors such as fluctuating demand, unpredictable lead times, and uncertainties of supplier performance poses formidable challenges to the dependability of supply chain systems (Hmioui, Bentalha & Alla, 2020).Recognizing the critical need for a comprehensive and adaptive approach, this research introduces a Enhancing Dependability in a Pharmaceutical Supply Chain".The research is concluded in "Conclusion and Future Work".

THE INTEGRATION OF DEPENDABILITY FACTORS
The incorporation of dependability factors in the proposed model, DROAIM, is a pivotal aspect that enhances its capability to address the intricacies of real-world supply chain operations.This deliberate inclusion acknowledges and accounts for the multifaceted challenges and uncertainties that characterize the dynamic business environment in which supply chains operate.
Demand for products and services within a supply chain system is seldom static.They fluctuate due to various factors such as seasonality, market trends, and unforeseen occurrences.By integrating demand fluctuations into a model, the DROAIM recognizes the need for adaptability in order allocation and inventory management processes (Nemtajela & Mbohwa, 2016).Hence, this enables the system to respond to changes in consumer behavior dynamically, preventing understock or overstock situations.
Lead time uncertainty refers to variations in the time for products to move through the supply chain, from order placement to delivery.The DROAIM considers this uncertainty, recognizing that delays or unexpected accelerations in lead times can substantially impact inventory levels and order fulfillment.By incorporating lead time variability, the model becomes more robust in managing the inherent uncertainties associated with logistics and transportation.
Supplier reliability is a critical factor influencing the overall dependability of a supply chain.The DROAIM explicitly integrates supplier performance metrics into its decisionmaking framework.This includes assessing the historical reliability of suppliers, assessing their capability to meet delivery deadlines, and considering the quality of goods provided.By factoring in supplier performance, the model promotes a proactive approach to mitigating risks associated with unreliable suppliers, ultimately contributing to the dependability of the entire supply chain (Eckhart et al., 2019;Erdogan et al., 2021).
Real-world supply chain scenarios are characterized by their complexity, often involving many interconnected processes, stakeholders, and external factors.The DROAIM acknowledges this complexity by simultaneously considering a range of dependability factors.The model's capability to handle the interplay among fluctuations in demand, lead time uncertainties, and variations in supplier performance reflects a holistic understanding of the challenges faced by supply chain practitioners.
The incorporation of dependability factors enables the DROAIM to make adaptive decisions.In response to changes in demand patterns, unexpected delays in lead times, or fluctuations in supplier performance, the model can dynamically adjust order allocation and inventory levels concurrently.Hence, this adaptability is crucial for maintaining an efficient and reliable supply chain, especially in industries where responsiveness to altering conditions is a key determinant of success.
By explicitly incorporating dependability factors such as demand fluctuations, lead time uncertainty, and supplier performance, the DROAIM demonstrates a nuanced understanding of the challenges inherent in real-world supply chain operations.This comprehensive approach appears to be a robust tool for practitioners seeking to improve the dependability and resilience of their supply chain systems in the face of dynamic and unpredictable business environments.

COMPREHENSIVE APPROACH
The DROAIM sets itself apart by comprehensively evaluating the key components' dependability within the supply chain system.This holistic assessment encompasses suppliers, transportation networks, and internal procedures, aiming to uncover vulnerabilities and strengths throughout the entire supply chain.
In the realm of supply chain management, the reliability and performance of suppliers play a pivotal role.The DROAIM employs a systematic approach to assess the dependability of suppliers.This involves assessing factors such as the consistency of product quality, adherence to delivery schedules, and responsiveness to fluctuations in demand.By quantifying and analyzing these aspects, the model provides a nuanced understanding of each supplier's contribution to the overall reliability of the supply chain (Hmioui, Bentalha & Alla, 2020).
Efficient transportation is critical for a smooth flow of goods within the supply chain.The DROAIM considers the dependability of transportation networks by assessing factors like transit times, route efficiency, and the reliability of carriers.The model takes into account the potential impact of uncertainties, such as weather conditions or unexpected disruptions, on the transportation system.This evaluation ensures that the supply chain is resilient in the face of external challenges, minimizing the risk of delays and disruptions (Ezzahra, Rachid & Roland, 2022;Riesco & Villagrá, 2019).
Internal processes and procedures are the backbone of any supply chain.The DROAIM scrutinizes the dependability of internal operations, including order processing, inventory management, and demand forecasting.By assessing the efficiency and robustness of these internal procedures, the model identifies areas for improvement and optimization.This internal evaluation is crucial for enhancing the overall dependability of the supply chain by addressing potential bottlenecks or inefficiencies within the organization.
The comprehensive assessment conducted by the DROAIM goes beyond individual assessments of suppliers, transportation networks, and internal procedures.It seeks to integrate these assessments into a unified understanding of the supply chain's dependability.By doing so, the model identifies vulnerabilities that may arise from dependencies or interactions between different components.Simultaneously, it recognizes strengths that contribute positively to the overall resilience and reliability of the supply chain.This holistic perspective enables decision-makers to prioritize improvement areas and strategically allocate resources to fortify weak links in the supply chain.
The DROAIM's evaluation of the dependability of suppliers, transportation networks, and internal procedures is a robust and detailed process.It provides a comprehensive understanding of the supply chain's strengths and weaknesses, empowering organizations to make informed decisions and implement targeted improvements for a more resilient and reliable supply chain operation.

METHODOLOGY
The model utilizes a stochastic model, reliability-based supplier evaluation, dynamic algorithm, and real-time analytics.This multifaceted approach ensures a robust and flexible framework that can adapt to altering requirements in the supply chain.

Stochastic model
Stochastic models are mathematical models that incorporate randomness or uncertainty.In supply chain management, stochastic models are employed to account for the inherent variability in demand, lead times, and other relevant factors.These models allow for the simulation of different scenarios, considering the probabilistic nature of certain events.
The specific novel stochastic model that was incorporated into the DROAIM is the Time-Varying Autoregressive (TVAR) (Haslbeck, Bringmann & Waldorp, 2021).This model is designed to capture the time-dependent and dynamic nature of supply chain processes, making it well-suited for applications in inventory management and order allocation.This model extends conventional autoregressive models by introducing timevarying coefficients, allowing it to adapt to changing conditions over time (Baptista de Souza, Kuhn & Seara, 2019).In the context of supply chain management, the TVAR was applied to represent the stochastic behavior of demand and lead times together, considering the evolving patterns and trends.
Specifically, the TVAR allows for coefficients to change over time, enabling it to capture shifts in demand patterns, seasonality, and other dynamic factors.It requires historical data for training and continuously updated data for real-time adaptation.Integration with real-time analytics in the DROAIM facilitates the flow of this data (Hong, Dai & Wang, 2016).The model automatically adjusts its parameters as new data becomes available, making it well-suited for dynamic supply chain environments with evolving demand and lead-time characteristics.Also, it incorporates market trends, economic indicators, and supplier performance metrics, providing a holistic view of supply chain dynamics.
A vector auto regression (VAR), presented in Eq. ( 1), is a statistical method implemented to represent a complex system consisting of multiple time series variables (Gholamzadeh & Bourbour, 2020).A VAR model of order k for the two variables X 1;t , and X 2;t comprises the subsequent pair of equations for every t (Magnant, Giremus & Grivel, 2015;Haslbeck, Bringmann & Waldorp, 2021): where the residual terms are vectorial white noise, meaning they exhibit random behavior delineated by Eq. (2).
It must be noted that the addition of the TVAR to the DROAIM makes stochastic modeling simpler and helps us understand how the supply chain changes over time more nuancedly.This, in turn, contributes to the overall dependability of the supply chain operations addressed by the DROAIM framework.

Reliability-based supplier evaluation
Reliability-based suppliers assess the dependability and performance of suppliers based on historical data and performance metrics.This evaluation can include factors such as ontime delivery, product quality, and responsiveness to changes in demand.Reliability metrics play a crucial role in evaluating the performance of suppliers and other components within the supply chain.Table 1 presents the reliability metrics that were incorporated into the DROAIM.
By incorporating the mentioned reliability metrics, the model ensures that supplier selection is not solely based on cost but also on the supplier's capability to meet commitments consistently.This promotes a more robust supply chain by reducing the risks associated with unreliable suppliers.

Dynamic algorithm
Dynamic algorithms can adapt and make real-time decisions based on altering conditions.In order to allocate and manage inventory, dynamic algorithms can optimize decisions as new information becomes available.This adaptability is crucial for responding promptly to unexpected events and maintaining optimal supply chain performance.
The novel dynamic algorithm implemented in the DROAIM is the Adaptive Reinforcement Learning algorithm that leverages reinforcement learning principles to dynamically adjust order allocation and inventory management strategies based on realtime feedback and altering conditions within the supply chain.
The model defines the state space, including current inventory levels, historical demand patterns, supplier reliability metrics, and real-time market conditions.This representation captures the dynamic nature of the supply chain.Also, it specifies the set of actions that the algorithm can employ, such as adjusting order quantities, reevaluating supplier priorities, or updating inventory policies.These actions are flexible to accommodate the varying demands and uncertainties in the supply chain.In addition, it develops a reward system that quantifies the performance of the supply chain based on predefined objectives.For example, rewards were assigned to meet demand on time, minimize stockouts, and optimize inventory costs.The reward system guides the algorithm towards learning strategies that align with the overarching goals of the DROAIM.
A deep Q-learning technique is implemented to enable the algorithm to learn optimal decision-making policies over time.Q-learning is a reinforcement learning technique that estimates the score of taking a particular action in a specific state.The The parameter e is employed in the exploration-exploitation trade-off.A higher e value indicates a greater inclination for the agent to engage in exploration and experiment with novel actions.Conversely, a lower e value suggests a higher tendency for the agent to utilize its available knowledge and pick actions with higher estimated Q-scores.The notation Q s; a ð Þ represents the present estimation of the anticipated future benefits for picking action a ð Þ in the state s ð Þ.The learning rate, denoted as a, governs the speed at which the Q-network adjusts its estimations.The variable r represents the immediate reward obtained after acting a ð Þ in the state b ð Þ.The variable c, known as the discount factor, defines the significance of future benefits s and a represent the subsequent state and action, respectively (Nai, Fang & Zhao, 2022;Soni, Vyas & Hiran, 2022).
The model also integrates exploration-exploitation strategies to balance the algorithm's exploration of new strategies with the exploitation of known optimal strategies.This ensures adaptability to altering conditions while maintaining efficiency in decision-making processes.Moreover, it allows the algorithm to update itself based on real-time feedback and performance metrics dynamically.This adaptability ensures that the algorithm can learn from and respond to evolving patterns in demand, lead times, and supplier reliability.Finally, the method connects the algorithm with real-time analytics to continuously feed it with the latest data.This integration enables the algorithm to make decisions based on the most up-to-date information, contributing to the overall responsiveness of the DROAIM.
The following is a pseudocode representation of the Adaptive Reinforcement Learning algorithm implemented in the DROAIM.

Real-time analytics
Real-time analytics involves the continuous analysis of data as it is generated, providing immediate insights and supporting timely decision-making (Dangal & Bloom, 2020;Morshed, Rana & Milrad, 2016).In supply chain management, real-time analytics are employed to surveil key performance indicators, track inventory levels, and pinpoint emerging trends (Puneeth Kumar, Manjunath & Hegadi, 2018).This real-time insight enables proactive decision-making processes, allowing the supply chain to quickly tune to alterations in demand, supplier performance, or other critical factors (Gong et al., 2020).
Combining these elements into the DROAIM creates a powerful and adaptable framework where the stochastic model provides a probabilistic understanding of demand and lead time uncertainties, and reliability-based supplier assessment ensures that supplier performance is considered in decision-making processes, enhancing the dependability of the overall supply chain.In addition, the dynamic algorithm enables the model to adapt to real-time changes in demand, supply chain conditions, and supplier performance.This algorithm continuously optimizes order allocation and inventory management strategies, ensuring the system remains responsive to evolving circumstances.Finally, real-time analytics provide immediate insights into the supply chain's current state, allowing for proactive decision-making processes, decreasing the impact of disruptions and enhancing overall supply chain resilience.

CASE STUDY AND SIMULATION: ENHANCING DEPENDABILITY IN A PHARMACEUTICAL SUPPLY CHAIN
The efficacy of the proposed model is validated through a case study and simulations employing synthetic data.This empirical testing provides practical insights into the performance of the DROAIM.
PHZ Pharmaceuticals, a global pharmaceutical company, operates in a highly regulated and dynamic industry.The company faces challenges related to fluctuations in demand for various medications, uncertainties in lead times, and the need for stringent quality control.PHZ Pharmaceuticals is exploring ways to improve the dependability of its supply chain operations.The company implements the DROAIM system, incorporating stochastic models, reliability-based supplier evaluation, dynamic algorithms, and real-time analytics.The primary objective is to evaluate the practical effectiveness of the DROAIM in improving the overall dependability of PHZ Pharmaceuticals' supply chain.Synthetic data will be implemented to simulate various scenarios, allowing for a comprehensive assessment of the model's performance.

Methodology
Synthetic data is generated to simulate demand fluctuations, lead time uncertainties, and supplier performance metrics (Fig. 2).The data is designed to mirror the characteristics of PHZ Pharmaceuticals' actual supply chain but with controlled variations to test the model under diverse conditions.
The DROAIM is implemented, integrating the proposed TVAR to capture demand uncertainties, reliability-based supplier assessment to assess supplier performance, dynamic algorithm Deep Q-learning network to adapt to altering conditions, and real-time analytics for immediate insights.
Various simulation scenarios represent distinct supply chain challenges, such as sudden demand increases, supplier performance disruptions, and unexpected lead times changes.These scenarios aim to stress-test (Fig. 3) the DROAIM under realistic conditions.Key performance metrics include supply chain dependability, stockout occurrences (Fig. 4), and inventory optimization.These metrics are utilized to assess the model's impact on supply chain performance quantitatively.
The performance of the DROAIM is compared with the baseline model (Fig. 5) currently employed by PHZ Pharmaceuticals.Specifically, the Economic Order Quantity (EOQ) model is a conventional method that determines the optimal order quantity to minimize total inventory costs, considering a fixed order quantity for each replenishment cycle.This model is a baseline for comparing the DROAIM's dynamic order allocation and inventory management.This comparison provides a benchmark for evaluating the relative improvement brought about by the proposed model.
Sensitivity analysis is conducted to understand how the DROAIM responds to variations in input parameters.This analysis helps pinpoint the proposed model's robustness and capability to adapt to distinct supply chain conditions (Fig. 6).

Outcomes and insights
The DROAIM demonstrates a substantial advancement in overall supply chain dependability compared to baseline models (Fig. 7).The proposed model effectively navigates through demand fluctuations and uncertainties in lead times, leading to a more reliable supply chain.
The simulation results show a noticeable reduction in stockout occurrences (Fig. 8), indicating the model's capability to optimize inventory levels and allocate orders efficiently.This contributes to improved customer satisfaction and revenue stability.
The DROAIM optimizes inventory levels (Fig. 9), ensuring PHZ Pharmaceuticals maintains adequate stock without excessive overstocking.This results in cost savings and enhances operational efficiency.The model exhibits a high adaptability to altering supply chain conditions through dynamic algorithms.It responds promptly to sudden increases in demand, disruptions in supplier performance, and other unforeseen events (Fig. 10).
The outcomes of the validation process provide actionable insights for PHZ Pharmaceuticals' decision-makers.Recommendations include strategies for proactive supplier management, dynamic order allocation, and continuous monitoring of key performance indicators.The case study, supported by synthetic data and empirical testing, contributes valuable knowledge to the field of supply chain management.It highlights the practical applicability of the DROAIM and its potential to address common challenges in  The outcomes provide actionable insights and contribute valuable knowledge to the broader field of supply chain management, reinforcing the model's potential for real-world applications.The findings highlight substantial enhancements in the supply chain system's overall dependability, reduced stockout occurrences, and optimal inventory levels.These results indicate the practical utility and effectiveness of the DROAIM in real-world scenarios.The model exhibits adaptability to various industry challenges, making it a versatile tool for supply chain practitioners.This adaptability is crucial in addressing the diverse and evolving challenges different businesses face.More up-to-date examples and

CONCLUSION AND FUTURE WORK
The research introduces an innovative decision-making model that incorporates dependability criteria into designing and optimizing order allocation and inventory management systems.The suggested model seeks to improve the overall dependability of supply chain operations by integrating elements such as fluctuations in demand, uncertainty in lead time, and the suppliers' performance.The efficacy of the suggested DROAIM has been empirically evaluated and demonstrated to substantially improve the reliability and trustworthiness of the supply chain system.The system, which incorporates probabilistic models, supplier evaluation based on reliability, algorithms that adjust in realtime, and analytics, has effectively minimized stockouts, optimized inventory levels, and enhanced flexibility in altering situations.This complete decision-making framework considers dependability variables, providing a more resilient and dependable way to allocate orders and manage inventory.Pharmaceutical decision-makers can utilize these findings to advance operational effectiveness and enhance consumer contentment.
While the current research provides a solid foundation for the integration of dependability factors into the supply chain decision-making process, several avenues for future research and improvement can be explored.These include real-world implementation with specific industry partners, fine-tuning parameters based on industry-specific characteristics, integrating advanced technologies like blockchain and artificial intelligence, involving suppliers more directly in decision-making processes, developing dynamic risk management strategies, examining its applicability across different industries with unique supply chain challenges, and assessing its long-term impact on supply chain performance,